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arXiv · 1809.04161

$C^{1,\alpha}$ Isometric Embeddings of Polar Caps

Camillo De Lellis·Dominik Inauen
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Abstract

We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the H\"older space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $\alpha > \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any $\alpha < \frac{1}{2}$ we construct $C^{1,\alpha}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.

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Camillo De Lellis, Dominik Inauen. 2018-09-11. $C^{1,\alpha}$ Isometric Embeddings of Polar Caps. https://arxiv.org/abs/1809.04161

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